Partial Differential Equations
The solution of ordinary differential equations with constant coefficients by means of the L-transformation is accomplished very easily, since the L-transformation removes the differentiation, a transcendental operation, and an algebraic image equation is obtained. When the original equation contains derivatives with respect to two variables, say x and t, that is it represents a partial differential equation, then the L-transformation applied to the variable t will remove differentiation with respect to t, and the image equation is an ordinary differential equation, with the variable x. Obviously, for this purpose we must presume that t varies in the interval 0 ≦ t < ∞ the variable x may range in an interval which may be bounded or unbounded at one or both sides. Accordingly, we have in the xt-plane (see Fig. 48) as fundamental region of the partial differential equation either a half-strip or a quadrant or a half-plane.
KeywordsPartial Differential Equation Ordinary Differential Equation Singular Point Asymptotic Expansion Image Space
Unable to display preview. Download preview PDF.